Motion of a scalar field coupled to a Yang Mills field reformulated locally with some gauge invariant variables

Latex2e, 20 pages, some typos corrected, journal ref addedThis paper exposes a reformulation of some gauge theories in terms of explicitly gauge-invariant variables. We show in the case of Scalar QED that the classical theory can be reformulated locally with some gauge invariant variables. We discuss the form of some realistic asymptotic solutions to these equations. The equations of motion are then also reformulated in the non-abelian case